Yuan, Jin; Budarina, Natalia; Dickinson, Detta On the number of polynomials with small discriminants in the Euclidean and \(p\)-adic metrics. (English) Zbl 1275.11109 Acta Math. Sin., Engl. Ser. 28, No. 3, 469-476 (2012). The main result of this paper is a lower bound for the number of polynomials with integer coefficients of bounded degree and height satisfying the property that their discriminant is ‘small’ and divisible by a large power of a fixed prime. The precise statement uses the following notation. Let \(P_n(Q)\) denote the set of non-zero polynomials with integer coefficients of degree \(\leq n\) and height \(\leq Q\). The height of a polynomial is defined as the maximum of the absolute values of its coefficients. Let \(D(P)\) denote the discriminant of a polynomial \(P\). For \(v\geq0\) let \[ \mathcal{P}_n(Q,v)=\{P\in P_n(Q):\;1\leq|D(P)|< Q^{2n-2-2v},\;|D(P)|_p<Q^{-2v}\}, \] where \(|\cdot|_p\) denotes the \(p\)-adic valuation. The authors prove that for \(n\geq3\) and \(0\leq v<1/3\) and all sufficiently large \(Q\) one has that \[ \#\mathcal{P}_n(Q,v)\geq C\,Q^{n+1-4v}. \] Reviewer: Victor Beresnevich (Heslington) Cited in 4 Documents MSC: 11J83 Metric theory 11J54 Small fractional parts of polynomials and generalizations 11J61 Approximation in non-Archimedean valuations Keywords:Diophantine approximation; discriminant; polynomial roots separation PDF BibTeX XML Cite \textit{J. Yuan} et al., Acta Math. Sin., Engl. Ser. 28, No. 3, 469--476 (2012; Zbl 1275.11109) Full Text: DOI References: [1] Mahler, K.: An inequality for the discriminant of a polynomial. Michigan Math. J., 11, 257–262 (1964) · Zbl 0135.01702 · doi:10.1307/mmj/1028999140 [2] Bernik, V., Götze, F., Kukso, O.: Lower bounds for the number of integral polynomials with given order of discrimants. Acta Arith., 133, 375–390 (2008) · Zbl 1222.11096 · doi:10.4064/aa133-4-6 [3] Bernik, V., Götze, F., Kukso, O.: On the divisibility of the discriminant of an integral polynomial by prime powers. Lith. Math. J., 48, 380–396 (2008) · Zbl 1228.11111 · doi:10.1007/s10986-008-9025-5 [4] Beresnevich, V., Bernik, V., Götze, F.: The distribution of close conjugate algebraic numbers. Compos. Math., 146, 1165–1179 (2010) · Zbl 1206.11091 · doi:10.1112/S0010437X10004860 [5] Bugeaud, Y., Dujella, A.: Root separation for irreducible integer polynomials. Bull. London Math. Soc., doi: 10.1112/blms/bdr085 (2011) · Zbl 1273.11049 [6] Bugeaud, Y., Mignotte, M.: Polynomial root separation. Intern. J. Number Theory, 6, 587–602 (2010) · Zbl 1205.11032 · doi:10.1142/S1793042110003083 [7] Evertse, J. H.: Distances between the conugates of an algebraic number. Publ. Math. Debrecen, 65, 323–340 (2004) · Zbl 1073.11066 [8] Schönhage, A.: Polynomial root separation examples. J. Symbolic Comput., 41, 1080–1090 (2006) · Zbl 1158.12300 · doi:10.1016/j.jsc.2006.06.003 [9] Zeludevich, F.: Simultane diophantische Approximationen abhangiger Grössen in mehreren Metriken. Acta Arith., 46, 285–296 (1986) [10] Bernik, V., Budarina, N., Dickinson, D.: Simultaneous Diophantine approximation in the real, complex and p-adic fields. Math. Proc. Camb. Phil. Soc., 149, 193–216 (2010) · Zbl 1221.11159 · doi:10.1017/S0305004110000162 [11] Sprindžuk, V.: Mahler’s problem in the metric theorem of numbers. Transl. Math. Monographs, Vol. 25, Amer. Math. Soc., Providence, RI, 1969 [12] Mohammadi, A., Salehi Golsefidy, A.: S-arithmetic Khintchine-type theorem. Geom. Funct. Anal., 19, 1147–1170 (2009) · Zbl 1272.11088 · doi:10.1007/s00039-009-0029-z This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.